Theorem 3.14 [1]}
Let \(\mu \)  be a Zariski-dense probability measure on a linear semisimple \(\mathbb {R}\) -split group \(H\)  and let \(V\)  be an irreducible algebraic representation of \(H\) . Then, for \(\mu ^{-\mathbb {N}^\ast }\) -a.e.  \(b=(b_{-1},\ldots )\) , any limit point \(\hat{\pi }_b\)  of the sequence \(\frac{b_{-1}\ldots b_{-n}}{\Vert  b_{-1}\ldots b_{-n}\Vert }\)  in \(\operatorname{End}(V)\)  has rank one and the same image. Moreover, for any hyperplane \(W<V\) , the set of \(b \in H^{-\mathbb {N}^\ast }\)  such that \(Im(\hat{\pi }_b) \in W\)  has zero measure.

